Volume 17, 2013
|Page(s)||531 - 549|
|Published online||03 June 2013|
Asymptotics of counts of small components in random structures and models of coagulation-fragmentation
Department of Mathematics, Technion-Israel Institute of
Revised: 18 September 2011
We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of components of fixed sizes in random structures. We then apply Schur’s tauberian lemma and some results from additive number theory and enumerative combinatorics in order to derive plausible sufficient conditions of convergence. Our results demonstrate that the common belief, that counts of components of fixed sizes in random structures become independent as the number of particles goes to infinity, is not true in general.
Mathematics Subject Classification: 60C05 / 60K35 / 05A16 / 82B05 / 11M45
Key words: Multiplicative measures on the set of partitions / random structures / coagulation-fragmentation processes / Schur’s lemma / models of ideal gas
© EDP Sciences, SMAI, 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.